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A sparse approximation of a function is an approximation given by a linear combination of “many” basis functions, where the vector of linear combination coefficients is sparse, i.e. it has only “a few” non-zero elements. Identifying a sparse approximation of an unkown function from a set of data can be useful for many applications in the automatic control field: system identification, basis function selection, regressor selection, nonlinear internal model control, nonlinear feed-forward control, direct inverse control, predictive control, fast online applications. In this paper, a combined ℓ1-relaxed-greedy algorithm for sparse identification is proposed and a Set Membership optimality analysis is carried out. Assuming that the noise affecting the data is bounded in norm and that the unknown function satisfies a mild regularity condition, it is shown that the algorithm provides an almost-optimal (in a worst-case sense) approximation of the unknown function. A simulation example is shown, related to direct-inverse control of a power kite used for high altitude wind energy conversion.